General-purpose sequential machine for solving boolean satisfiability (SAT) problems in linear time

ABSTRACT

The invention is a general-purpose sequential machine for solving boolean satisfiability (SAT) problems for functions of n variables and m clauses in linear time with complexity O(m), independent of the number of variables in the function. With current hardware technology, a value of n=32 variables can be achieved. The machine can serve as a basic building block to develop faster SAT solvers.

1. TECHNICAL FIELD

[0001] The present invention relates in general to the field of computing, more specifically to a hardware-based method for solving the boolean satisfiability problem (SAT) in linear time, independent of the number of variables, for any n-variable instance of the problem, where the maximum value of n is set by hardware limitations.

2. BACKGROUND

[0002] The boolean satisfiability problem (or SAT) is a well-known problem, which belongs to the class of NP-complete problems [2]. SAT is stated as follows: given a boolean formula F(x₁,x₂, . . . x_(n)), find if there exists an assignment of binary values to each (x₁,x₂, . . . x_(n)), such that F equals 1. So far, no one has found a polynomial-time solution to SAT; finding one would imply that NP-complete problems can be solved in polynomial time, which has not been proved yet. While most attempts have been tried in software [3], in recent years, hardware solutions have been attempted for SAT [1,4,5,6]. However, such solutions are tailored for specific problem instances, instead of a general-purpose solution. Also, they don't have a guaranteed linear complexity as a function of only the number of clauses.

3. BRIEF SUMMARY

[0003] The invention is a general-purpose sequential machine for solving the SAT problem (GP-SAT). The machine can solve any n-variable instance of the SAT problem in O(m) time, where m is the number of clauses, independent of the number of variables, n, where the maximum value of n is set by hardware limitations. With current hardware technology, a value of n=32 variables can be achieved. Although this number might be considered small, the machine can serve as a basic building block to develop faster SAT solvers.

4. BRIEF DESCRIPTION OF THE DRAWINGS

[0004]FIG. 1. Schematic diagram of GP-SAT

[0005]FIG. 2. Truth table for SAT Decoder

[0006]FIG. 3. Truth table for SAT Decoder, n=3

[0007]FIG. 4. Schematic diagram of the R-S Flip-Flop Array

5. DETAILED DESCRIPTION 5.1. Restatement of SAT

[0008] An n-variable boolean formula F(x₁,x₂, . . . x_(n)) with m clauses, can be expressed in Conjunctive Normal Form (CNF) as a product of clauses, as follows: $F = {\prod\limits_{i = 1}^{m}{Ci}}$

[0009] where Π represents the conjunction (AND) operation, and each Ci is a clause Each Ci can be defined as a function Ci(x₁,x₂, . . . x_(n)), and can be represented by a sum (OR) of variables, as follows: ${Ci} = {\sum\limits_{j = 1}^{n}{Lij}}$

[0010] where each Lij is a literal, whose value is either x_(j), −x_(j), or 0. The value 0 indicates that the variable x_(j) is not in the clause, in which case the term does not have any effect.

[0011] Since F is an AND, F is satisfiable if there exists an assignment that makes all Ci to evaluate to 1. Conversely, F is unsatisfiable if for all assignments, at least one Ci evaluates to 0.

[0012] We define the coverage set Vi of a clause Ci as the set of assignments for which Ci evaluates to 0. The coverage set V of F is defined as the union: $V = {\bigcup\limits_{i = 1}^{m}V_{i}}$

[0013] V contains all the assignments for which at least one of the Ci will evaluate to 0. Clearly, if |V|=2^(n), then F is unsatisfiable, since at least one Ci will evaluate to 0, for each possible assignment. Otherwise, F is satisfiable.

5.2. The GP-SAT Machine

[0014] So far, nobody has found a way of building in polinomial time the set V in a single-processor machine (otherwise, we can conclude that P=NP). However, combinatorial logic can be used to generate in parallel all elements of each Vi in V in a single clock cycle, as explained in this section.

[0015] A schematic diagram of the GP-SAT machine for n variables is presented in. The machine has 2n+2 inputs: two inputs (Lj and Xj) for each variable, a RESET input and a CLK (clock) input. Each Ci clause is input before the rising edge of the CLK input. For each Ci, the inputs Lj and Xj are set according to the following truth table: TABLE 1 Truth table for assignment of Xj and Lj Condition for variable x_(j) Xj Lj Appears in the clause, non-negated 0 0 Appears in the clause, negated 0 1 Does not appear in the clause 1 X

[0016] Formulas are processed in the machine as follows:

[0017] 1. Set and unset RESET to clear R-S memory.

[0018] 2. For each Ci in F:

[0019] a. Input clause Ci′ according to Table 1.

[0020] b. Raise and lower CLK

[0021] 3. Look at output SAT

[0022] The SAT Decoder is a special decoder with 2n inputs and 2^(n) outputs. It is responsible for identifying the coverage set of each clause. Each output Ym corresponds to a variable assignment, where Y0 corresponds to an assignment of all 0s, and Y₂ ^(k-1) corresponds to an assignment of all 1s . If all X in the input are 0, meaning that all variables are present in the clause, the decoder behaves as a normal decoder. Otherwise, for each absent variable in the clause, all outputs corresponding to elements of the coverage set are set to 1. A truth table for the SAT Decoder is presented in FIG. 2. As an example, a truth table for a SAT decoder for n=3 is presented in FIG. 3.

[0023] Outputs from the decoder are fed into an R-S Flip-Flop Array. This array serves as a memory for building the coverage set. A schematic diagram of the array is presented in FIG. 4.

[0024] Finally, all outputs from the flip-flop array are fed into a NAND circuit, which gives the SAT output. When the coverage set reaches the maximum of 2^(n), all inputs of the NAND are set to 1, causing the output SAT to be 0.

7. References

[0025] [1] Abramovici, M. and Saab, D., “Satisfiability on Reconfigurable Hardware”, in 7^(th) International Workshop on Field Programmable Logic and Applications (1997).

[0026] [2] Cook, S., “The complexity of theorem proving procedures”, Proceedings of the 3rd Annual ACM Symposium on the Theory of Computation (1971) 151-158.

[0027] [3] Du, D., Gu, J. and Pardalos, P. M., “Satisfiability Problem: Theory and Applications”, American Mathematics Society (1996).

[0028] [4] Suyama, T., Yokoo, M., and Sawada, H., “Solving Satisfiability on FPGAs” , in 6^(th) International Workshop on Field-Programmable Logic and Applications (1996).

[0029] [5] Zhong, P., Martonosi, M., Ashar, P., and Malik, S., “Solving Boolean Satisfiability with Dynamic Hardware Configurations” , International Workshop Field-Programmable Logic and Applications (FPL'98), pp. 326-335.

[0030] [6] Zhong, P., Martonosi, M., Ashar, P., and Malik, S., “Accelerating Boolean Satisfiability with Configurable Hardware”, in FCCCM'98 (1998). 

What is claimed is:
 1. A sequential machine that solves SAT for any boolean formula of n variables and m clauses, in linear time O(m), which is independent of n; where the maximum value of n is constrained by hardware limitations.
 2. The machine of claim 1, wherein each literal of each clause Ci of a boolean function F input to the machine is represented by two bits; namely the Xij bit to indicate the presence of the jth variable in the clause; and the Lij to indicate whether the variable is negated or not in the literal (if the literal is present in the clause).
 3. The machine of claim 2, wherein the following process is used to solve SAT for a function F: a. Set and unset RESET to clear memory b. For each clause Ci in F: i. Input clause Ci ii. Raise and lower CLK c. Look at output SAT, where SAT=1 means satisfiable, and SAT=0 means unsatisfiable
 4. The machine of claim 3, wherein a decoder is tailored to compute, in a single clock cycle, the set of assignments that make the input clause evaluate to a binary 0 (or coverage set).
 5. The machine of claim 4, wherein the output of the decoder is fed into an R-S flip-flop array to save the state of previous computations and obtain the union of all coverage sets.
 6. The machine of claim 5, wherein the output of the R-S flip-flop array is input to a NAND gate to produce the output SAT. 